Chances Are (43 page)

War in Virginia became a repeated game with a stable solution. Lee would win the local advantage, but that larger Union army would just stretch a little farther out and press a little farther forward. Grant lost more than 60 percent of his force during the campaign: at Cold Harbor, his soldiers pinned notes with their names and addresses to their uniforms, certain they were marching to death. But Grant's losses were replenished; Lee lost more than a third of his experienced generals—and these were irreplaceable. The end took long in coming, but the nature of equilibrium made it inevitable: there was no better strategy available to Lee, given the strategy of Grant.

The contrast between McClellan and Grant shows how what you value influences what happens to you. The payoff from your matrix depends on the assumptions you bring to the game—and there may well be things you value beyond simple victory or fear beyond defeat. Thermopylae, despite the loss of their king and finest fighters, was a triumph for the Spartans because it glorified their sense of fortitude in despair; Dunkirk was a triumph for the English belief in civilian decency and muddling through. Persians and Germans alike found their opponents' view of these obvious defeats entirely baffling.

 

Grant said that he must always assume that General Lee would do what was best. Game theory presupposes a rational player—that is, someone who will choose the strategy that best supports his utility and who, when faced with a choice of equilibria (since there can be more than one Nash equilibrium in a game), will choose the more desirable. What happens, though, when an opponent is
not
rational—when he chooses the stable but least desirable quadrant of the matrix?

In 1854, Paraguay passed into the hands of Francisco Solano López. Plump, flattered to be told he resembled Bonaparte, he felt destiny strong upon him. In 1865, López provoked simultaneous wars with Argentina and Brazil; Uruguay joined a Triple Alliance against him. What followed was one of the bloodiest conflicts on record.

López's army was gone—killed, wounded or captured—within eighteen months. Every male Paraguayan had been conscripted: ten-year-olds fought and died beside their grandfathers. The new armies marched half-naked, their colonels barefoot. Naval infantry units attacked Brazilian ironclads armed only with machetes. As the allies advanced, López's paranoia grew: he tortured and killed most of his government, the civil service, five hundred members of the foreign diplomatic corps, and two of his own brothers. And yet Paraguayans continued to fight for him with suicidal bravery. Finally, he was cornered by Brazilian troops on the banks of the Aquidaban and shot as he attempted to swim to freedom.

López's last words were reported as: “I die for my country”—but it would be more accurate to say his country died for him. Paraguay lost 90 percent of its male population; for years afterward, polygamy was tolerated. By any reasonable standard, the choice López made was senseless—yet a whole nation followed him into the abyss. The power of war binds individually rational judgments together in irrationality.

 

Sometimes you play the game; sometimes the game plays you: situations that are themselves irrational but stable need no paranoid dictator to set them going. Mark Shubik (another RAND staffer) described a particularly worrying party game. He would offer to auction a dollar bill to the highest bidder; the only difference from a traditional auction was that the second-highest bidder would also have to pay. So if you bid 70 cents for the dollar and your neighbor bids 75, he gains a quarter while you lose your money—and also lose face. Even if you have to buy the dollar for $1.10, at least you've lost only a dime; the underbidder has lost much more. The bidding would usually slow as it approached the dollar mark, but once past would zoom well beyond it. People were buying a dollar for an average price of $3.40, just to avoid being the person who had bid so much for nothing. It's sometimes called the Macbeth principle (“I am in blood step't so far that, should I wade no more, returning were as tedious as go o'er”), but it applies to much more than just auctions or political assassination—it describes engineering white elephants, strikes, dead-end weapons development, and all the little conflicts that escalate relentlessly.

McGeorge Bundy, who had been Kennedy's and Johnson's security and Indochina man, once visited a Boston secondary school during the early seventies, at the height of the protests against America's involvement in Vietnam. It was not a welcoming audience: the young, earnest faces surrounding him glowed with righteous disdain for the compromised warmonger. In a quiet voice, Bundy began: “I'll take you through the events as they happened, starting in 1945; when you hear me come to the place where we should have stopped, raise your hand.” He started with simple, innocent matters: helping a damaged British navy, bolstering a weak France, supporting a newly independent friendly country, shoring up a local army—a policy here, a commitment there . . . penny bids. Each further step seemed no more than a logical way to protect the position already established—and there was already so much to lose. The audience nodded; the first hand did not go up until Bundy had reached the point of full commitment of regular troops: hot war. The students, like the U.S. government, had bought the dollar several times over.

 

In retrospect, irrational decisions by rational people are often revealed to be products of time and ignorance. Of all the simplifications built into early game theory, the most questionable was
perfect knowledge:
having the various payoffs for each side right there in front of you at the beginning, marked in the matrix. In reality, many conflicts are shaped more like trees than matrices: the outcomes lie out on distant branches that depend for their existence on a sequence of contingent decisions. Trying to figure your chances so far into the future may convince you that “you can't get there from here”: you may miss the most desirable payoff and choose an initial strategy that eventually goes against your interests—acting, in error, like an irrational player. Game theory calls this
the trembling hand:
the likelihood that previously closed realms of probability, favorable or unfavorable, can open up through local irrationality.

The system of Great Power alliances in early 1914 resembled one of those circus acts where the whole family, playing musical instruments, perch in an unlikely pyramid on Papa, who balances, in turn, on a ball. The ball in this case was Serbia, guaranteed by Orthodox Russia (supported by France) and menaced by Catholic Austria (supported by Germany). Britain, wary of Continental adventures, had only a general understanding with France and a general suspicion of Germany; but, having helped invent Belgium, had guaranteed its neutrality—and the German battle plan required the violation of that neutrality in the interests of maneuver. So everything, the future hopes of millions, was poised on one simple question: could Serbia refrain from provoking Austria?

The Austrian heir apparent, Archduke Franz Ferdinand, was shot in Sarajevo; the hand of Serbian army intelligence was apparent in the deed. Austria's Foreign Minister, Count Berchtold, a dimwitted but devious bully, was more than willing to deceive his own colleagues and allies in his cold desire to punish his smaller neighbor. The ultimatum Berchtold issued to Serbia was deliberately designed to affront national honor—bait for the trap of war. Serbian politicians, having jammed their only typewriter, scribbled right up to the deadline, seeking a formula that would appear concessionary to Austria but not to their fellow Serbs. Behind the frenzy was a growing fatalism: “Ah, well,” said Jovanovic, the public information minister, “I suppose there's nothing to do but die fighting.”

Of course there was an alternative—it simply was too painful to consider. For these beleaguered ministers, it
seemed
more rational to set the fatal machine in motion than to submit. Linked mobilizations went ahead across Europe. Within three days, Russia, Germany and France were officially at war. At one moment Kaiser Wilhelm lost his nerve and tried to halt the relentless plan—but his commanding general explained, in tears, that the train schedule was too complex to meddle with now. The German Chancellor Bethman-Hollweg prayed: “When the iron dice roll, may God help us.” But God refused to play; instead, the millions fell beneath the trembling hand.

It's not always easy to see the better strategy several steps ahead: even if each choice is only between two actions, the universe of possible outcomes increases, at every step, as the power of 2. Seven simple decisions away from this moment, there are 128 different ways of imagining the future. That is why game theory has imported the idea of
commitment:
a deliberate, visible lopping away of branches from the tree of possibilities. When Cortez burned the boats with which he had crossed the Atlantic, both his men on the beach and the Aztecs spying from the bluffs knew he had only one plan: to conquer. Herman Kahn, archpriest of Cold War absolutism, worried that if a country under nuclear attack could choose
not
to retaliate, this would in itself make a first strike more imaginable. So he devised and tested the idea of the
Doomsday machine:
a system, placed deliberately beyond the government's control, that would retaliate automatically when it detected an enemy attack. The engineering report said this was “technically feasible”—always one of the most frightening phrases to emerge from the mouth of an American.

Game theory tips your hand, because it allows your opponent to assume you will make the rational choice—so if you don't want to reveal your choice, you may have to commit to appearing irrational. This is what Richard Nixon approvingly called the “Mad President” strategy—which, if it didn't frighten the enemy, certainly frightened us.

 

A crucial innovation in von Neumann's and Morgenstern's work was its way of representing utility. Before, preference had been relative: “I prefer this to that, and that to the other.” It was an ordinal function: it had no cardinal value, no number attached to it. The breakthrough in
The Theory of Games and Economic Behavior
was finding a formula to convert ordinal utility to cardinal, so the degree to which you desired an outcome could be plugged into the calculation of the probability of that outcome. This adds a new layer of sophistication to game theory, since different strategies often have different intrinsic probabilities of success as well as different payoffs if they succeed. Knowing these probabilities is the basis of the mixed strategy: you decide between available choices randomly, but with the random choice generator weighted according to the likelihood of success.

Throughout the winter of 1943-44, the Allies were preparing to invade France—and the Germans knew it. The alternative landing sites were strictly limited and occurred naturally to every mind that considered the problem: Pas de Calais or Normandy? The area around Calais offered a shorter sea journey, a smoothly shelving coast, and several excellent harbors. Normandy represented an overnight trip through rough sea to beaches with steep cliffs behind them and no natural shelter for landing craft or supply vessels. The Pas de Calais leads the invading army directly toward the flat, familiar fighting territory of Flanders and Picardy, with the prospect of a quick crossing of the great north-south river systems. The Norman beaches lead into Normandy itself—a dense checkerboard of tiny fields and impenetrable hedges. Probability favored the Pas de Calais, then—but this was equally obvious to the defenders, and the bloody failure of the Dieppe raid in 1942 had shown how hard it would be to attack a well-defended port. So both sides revolved the choice: take the high probability or the low? The whole question seemed to come down to the flip of a lopsided coin.

That is, indeed, how game theory would present the problem; yet, intuitively, this approach seems to depend on the game's being repeatable. Professional poker players bluff about as often as an optimal mixed strategy would suggest—but that's because they play game after game. What if everything depended on one hand? Could you really justify taking the low-probability route on the basis of a random choice?

Equations combine variables and constants. In scientific experiments you control only the variables, but in human affairs the constants may also be open to attack. The Allies chose the Normandy invasion route despite (or, perhaps, because of) its lower probability of success—this was the apparent constant. They then set about adjusting that constant, while at the same time keeping it unchanged in the minds of the defenders.

It was, of course, impossible to hide the invasion forces gathering on the coast opposite Normandy, but it was not impossible to disguise their relative importance. A huge phantom army, the “First U.S. Army Group,” was created—complete with canvas landing craft and inflatable tanks—and billeted in southeast England, convenient to the Pas de Calais. Normandy's lack of secure anchorage and difficulty of supply were brilliantly overcome by technology: the floating concrete Mulberry harbors and PLUTO, the first underwater fuel pipeline. Counterintelligence adroitly finessed German expectations: the double agent GARBO (Juan Pujol) warned Berlin of the Normandy invasions hours before they began—that is, just a little too late. He then exploited the credibility this warning gained him to convince the defenders that Normandy was merely a diversion and the real attack would be at the obvious point: the Pas de Calais. Twenty-one German divisions were kept out of the battle for two months in expectation of an invasion that never came.

 

How far have we come from Earl Waldegrave's card table? In many ways, we have never left it. Every negotiation, from the divorce courts to the United Nations, still involves accepting the least bad as a substitute for the best. Every leader, from parent to president, knows that leadership often simply means willingness to make the random choice required by a mixed strategy. If the winning choice were always determinable, we could leave government, like chess, to the computers.